Scotland Yard has decided not to investigate the expenses claims of the Conservative party co-chair, Lady Warsi, and has passed the matter back to the House of Lords, saying it has "determined that this is not a matter for the police".

The decision was announced as the Lords' standards commissioner launched a formal investigation into the cabinet minister's expenses.

Labour's call for a criminal investigation into the affair has thus been ruled out, and "as such the matter has been referred back to the House of Lords commissioner for standards," a Met spokesman said on Wednesday, adding: "The matter was considered in accordance with previously adopted procedures."

Warsi referred herself to the standards commissioner last week following allegations that she had failed to declare income to the Lords register of interests and that she had claimed up to £2,000 in accommodation expenses when living in a friend's flat in London for free. She has denied any wrongdoing, insisting that she had acted within the rules at all times.

She said she made an "appropriate payment" to her friend – Tory official Naweed Khan, who is now one of her aides – for the nights she stayed at a property in Acton, west London. But the property's owner, GP and former Conservative donor Wafik Moustafa, denies receiving any income from either Warsi or Khan during the time of her stay in 2008. At the time, she was claiming Lords subsistence of £165.50 a night.

Following an initial assessment, a spokesman for the Lords standards commissioner said he would hold a formal investigation.

Warsi wrote a letter of apology to Cameron over that matter, saying she was "sincerely sorry" for the embarrassment to the government. She said their relationship was well known to her private office, adding that she did not know she needed to declare they had "a common business interest as minority shareholders in a small food company".

She wrote: "I sincerely regret that I did not consider the significance of this relationship with Mr Hussain when the arrangements for the visit were being made. In retrospect, I accept that I should have made officials aware of the business relationship between Mr Hussain and myself, and for this I am sorry.

"I regret that this failure may have caused embarrassment to the government."

The prime minister said he accepted her apology but was asking Allan to "consider the issues that have been raised with respect to the ministerial code and to provide advice to me as rapidly as possible". He wrote: "There are clearly some lessons for future handling."

Cameron has been accused by Labour of being inconsistent for referring Warsi to the independent adviser on ministerial interest while refusing to take the same step with his culture secretary, Jeremy Hunt, who has also faced criticism over his conduct in office in relation to News Corp's BSkyB bid. Cameron insisted on Tuesday that they were two "very different cases".